There is actually another way. You can downgrade to an earlier version of Flash that still uses the /tmp. It ain't super easy, but there is an Adobe site for developers that has the old Flash versions-- http://kb2.adobe.com/cps/142/tn_14266.html
Download the file (126MB) Flash Player 10.1.102.64 and 9.0.2890 (that is all ONE file)
Extract the Zip file and then drill down the folder extracted to the tarball for Linux flashplayer10_1r102_64_linux.tar.gz
Inside that tarball is the libflashplayer.so
Navigate to /opt and open the /opt/mint-flashplugin-10.3 folder as root. Inside that folder, rename the libflashplayer.so to OLDlibflashplayer.so and then copy the libflashplayer.so that you extracted from your tarball into the folder.
That should make Flash use the /tmp as storage instead of the cache. Does on my machine, anyway, at least for the time being.
The reason I downgraded is that the VideoDownloadHelper is too slow for me.